Properties

 Label 45T26 Degree $45$ Order $180$ Cyclic no Abelian no Solvable yes Primitive no $p$-group no Group: $C_3^2:(C_5:C_4)$

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Group action invariants

 Degree $n$: $45$ Transitive number $t$: $26$ Group: $C_3^2:(C_5:C_4)$ Parity: $1$ Primitive: no Nilpotency class: $-1$ (not nilpotent) $|\Aut(F/K)|$: $1$ Generators: (1,39,12,5,38,11,4,37,15,3,36,14,2,40,13)(6,29,17,10,28,16,9,27,20,8,26,19,7,30,18)(21,44,32,25,43,31,24,42,35,23,41,34,22,45,33), (1,36,6,16)(2,40,7,20)(3,39,8,19)(4,38,9,18)(5,37,10,17)(11,21,26,31)(12,25,27,35)(13,24,28,34)(14,23,29,33)(15,22,30,32)(42,45)(43,44)

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$4$:  $C_4$
$10$:  $D_{5}$
$36$:  $C_3^2:C_4$

Resolvents shown for degrees $\leq 10$

Subfields

Degree 3: None

Degree 5: $D_{5}$

Degree 9: $C_3^2:C_4$

Degree 15: None

Low degree siblings

There are no siblings with degree $\leq 10$
Data on whether or not a number field with this Galois group has arithmetically equivalent fields has not been computed.

Conjugacy classes

 Cycle Type Size Order Representative $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1$ $9$ $2$ $( 6,41)( 7,42)( 8,43)( 9,44)(10,45)(11,36)(12,37)(13,38)(14,39)(15,40)(16,31) (17,32)(18,33)(19,34)(20,35)(21,26)(22,27)(23,28)(24,29)(25,30)$ $4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 2, 2, 1$ $45$ $4$ $( 2, 5)( 3, 4)( 6,21,41,26)( 7,25,42,30)( 8,24,43,29)( 9,23,44,28) (10,22,45,27)(11,16,36,31)(12,20,37,35)(13,19,38,34)(14,18,39,33)(15,17,40,32)$ $4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 2, 2, 1$ $45$ $4$ $( 2, 5)( 3, 4)( 6,26,41,21)( 7,30,42,25)( 8,29,43,24)( 9,28,44,23) (10,27,45,22)(11,31,36,16)(12,35,37,20)(13,34,38,19)(14,33,39,18)(15,32,40,17)$ $5, 5, 5, 5, 5, 5, 5, 5, 5$ $2$ $5$ $( 1, 2, 3, 4, 5)( 6, 7, 8, 9,10)(11,12,13,14,15)(16,17,18,19,20) (21,22,23,24,25)(26,27,28,29,30)(31,32,33,34,35)(36,37,38,39,40) (41,42,43,44,45)$ $10, 10, 10, 10, 5$ $18$ $10$ $( 1, 2, 3, 4, 5)( 6,42, 8,44,10,41, 7,43, 9,45)(11,37,13,39,15,36,12,38,14,40) (16,32,18,34,20,31,17,33,19,35)(21,27,23,29,25,26,22,28,24,30)$ $5, 5, 5, 5, 5, 5, 5, 5, 5$ $2$ $5$ $( 1, 3, 5, 2, 4)( 6, 8,10, 7, 9)(11,13,15,12,14)(16,18,20,17,19) (21,23,25,22,24)(26,28,30,27,29)(31,33,35,32,34)(36,38,40,37,39) (41,43,45,42,44)$ $10, 10, 10, 10, 5$ $18$ $10$ $( 1, 3, 5, 2, 4)( 6,43,10,42, 9,41, 8,45, 7,44)(11,38,15,37,14,36,13,40,12,39) (16,33,20,32,19,31,18,35,17,34)(21,28,25,27,24,26,23,30,22,29)$ $3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3$ $4$ $3$ $( 1, 6,41)( 2, 7,42)( 3, 8,43)( 4, 9,44)( 5,10,45)(11,16,21)(12,17,22) (13,18,23)(14,19,24)(15,20,25)(26,31,36)(27,32,37)(28,33,38)(29,34,39) (30,35,40)$ $15, 15, 15$ $4$ $15$ $( 1, 7,43, 4,10,41, 2, 8,44, 5, 6,42, 3, 9,45)(11,17,23,14,20,21,12,18,24,15, 16,22,13,19,25)(26,32,38,29,35,36,27,33,39,30,31,37,28,34,40)$ $15, 15, 15$ $4$ $15$ $( 1, 8,45, 2, 9,41, 3,10,42, 4, 6,43, 5, 7,44)(11,18,25,12,19,21,13,20,22,14, 16,23,15,17,24)(26,33,40,27,34,36,28,35,37,29,31,38,30,32,39)$ $15, 15, 15$ $4$ $15$ $( 1, 9,42, 5, 8,41, 4, 7,45, 3, 6,44, 2,10,43)(11,19,22,15,18,21,14,17,25,13, 16,24,12,20,23)(26,34,37,30,33,36,29,32,40,28,31,39,27,35,38)$ $15, 15, 15$ $4$ $15$ $( 1,10,44, 3, 7,41, 5, 9,43, 2, 6,45, 4, 8,42)(11,20,24,13,17,21,15,19,23,12, 16,25,14,18,22)(26,35,39,28,32,36,30,34,38,27,31,40,29,33,37)$ $3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3$ $4$ $3$ $( 1,11,36)( 2,12,37)( 3,13,38)( 4,14,39)( 5,15,40)( 6,16,26)( 7,17,27) ( 8,18,28)( 9,19,29)(10,20,30)(21,31,41)(22,32,42)(23,33,43)(24,34,44) (25,35,45)$ $15, 15, 15$ $4$ $15$ $( 1,12,38, 4,15,36, 2,13,39, 5,11,37, 3,14,40)( 6,17,28, 9,20,26, 7,18,29,10, 16,27, 8,19,30)(21,32,43,24,35,41,22,33,44,25,31,42,23,34,45)$ $15, 15, 15$ $4$ $15$ $( 1,13,40, 2,14,36, 3,15,37, 4,11,38, 5,12,39)( 6,18,30, 7,19,26, 8,20,27, 9, 16,28,10,17,29)(21,33,45,22,34,41,23,35,42,24,31,43,25,32,44)$ $15, 15, 15$ $4$ $15$ $( 1,14,37, 5,13,36, 4,12,40, 3,11,39, 2,15,38)( 6,19,27,10,18,26, 9,17,30, 8, 16,29, 7,20,28)(21,34,42,25,33,41,24,32,45,23,31,44,22,35,43)$ $15, 15, 15$ $4$ $15$ $( 1,15,39, 3,12,36, 5,14,38, 2,11,40, 4,13,37)( 6,20,29, 8,17,26,10,19,28, 7, 16,30, 9,18,27)(21,35,44,23,32,41,25,34,43,22,31,45,24,33,42)$

Group invariants

 Order: $180=2^{2} \cdot 3^{2} \cdot 5$ Cyclic: no Abelian: no Solvable: yes GAP id: [180, 24]
 Character table:  2 2 2 2 2 1 1 1 1 . . . . . . . . . . 3 2 . . . 2 . 2 . 2 2 2 2 2 2 2 2 2 2 5 1 1 . . 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1a 2a 4a 4b 5a 10a 5b 10b 3a 15a 15b 15c 15d 3b 15e 15f 15g 15h 2P 1a 1a 2a 2a 5b 5b 5a 5a 3a 15b 15d 15a 15c 3b 15f 15h 15e 15g 3P 1a 2a 4b 4a 5b 10b 5a 10a 1a 5b 5a 5a 5b 1a 5b 5a 5a 5b 5P 1a 2a 4a 4b 1a 2a 1a 2a 3a 3a 3a 3a 3a 3b 3b 3b 3b 3b 7P 1a 2a 4b 4a 5b 10b 5a 10a 3a 15b 15d 15a 15c 3b 15f 15h 15e 15g 11P 1a 2a 4b 4a 5a 10a 5b 10b 3a 15a 15b 15c 15d 3b 15e 15f 15g 15h 13P 1a 2a 4a 4b 5b 10b 5a 10a 3a 15c 15a 15d 15b 3b 15g 15e 15h 15f X.1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 X.2 1 1 -1 -1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 X.3 1 -1 A -A 1 -1 1 -1 1 1 1 1 1 1 1 1 1 1 X.4 1 -1 -A A 1 -1 1 -1 1 1 1 1 1 1 1 1 1 1 X.5 2 -2 . . B -B *B -*B 2 B *B *B B 2 B *B *B B X.6 2 -2 . . *B -*B B -B 2 *B B B *B 2 *B B B *B X.7 2 2 . . B B *B *B 2 B *B *B B 2 B *B *B B X.8 2 2 . . *B *B B B 2 *B B B *B 2 *B B B *B X.9 4 . . . 4 . 4 . -2 -2 -2 -2 -2 1 1 1 1 1 X.10 4 . . . 4 . 4 . 1 1 1 1 1 -2 -2 -2 -2 -2 X.11 4 . . . C . *C . -2 -B -*B -*B -B 1 D /E E /D X.12 4 . . . C . *C . -2 -B -*B -*B -B 1 /D E /E D X.13 4 . . . *C . C . -2 -*B -B -B -*B 1 E D /D /E X.14 4 . . . *C . C . -2 -*B -B -B -*B 1 /E /D D E X.15 4 . . . C . *C . 1 D /E E /D -2 -B -*B -*B -B X.16 4 . . . C . *C . 1 /D E /E D -2 -B -*B -*B -B X.17 4 . . . *C . C . 1 E D /D /E -2 -*B -B -B -*B X.18 4 . . . *C . C . 1 /E /D D E -2 -*B -B -B -*B A = -E(4) = -Sqrt(-1) = -i B = E(5)^2+E(5)^3 = (-1-Sqrt(5))/2 = -1-b5 C = 2*E(5)^2+2*E(5)^3 = -1-Sqrt(5) = -1-r5 D = -E(5)^2+2*E(5)^3 E = -E(5)+2*E(5)^4